Intersection of Quadrics in \(\mathbb {C}^n\), Moment-Angle Manifolds, Complex Manifolds and Convex Polytopes

  • Alberto VerjovskyEmail author
Part of the Lecture Notes in Mathematics book series (LNM, volume 2246)


These are notes for the CIME school on Complex non-Kähler geometry from July 9th to July 13th of 2018 in Cetraro, Italy. It is an overview of different properties of a class of non-Kähler compact complex manifolds called LVMB manifolds, obtained as the Hausdorff space of leaves of systems of commuting complex linear equations in an open set in complex projective space \({\mathbb P}_{\mathbb {C}}^{n-1}\).



I would like to thank Yadira Barreto, Santiago López de Medrano, Ernesto Lupercio and Laurent Meersseman for their suggestions, lecture notes, and sharing their thoughts about the different aspects of the subject of these notes with me during several years. I would like to thank also the referee for pointing several typos and important mathematical details.

This work was partially supported by project IN106817, PAPIIT, DGAPA, Universidad Nacional Autónoma de México.


  1. [AK01]
    Yu. Abe, K. Kopfermann, Toroidal Groups: Line Bundles, Cohomology and Quasi-Abelian Varieties. Lecture Notes in Mathematics, vol. 1759 (Springer, Berlin, 2001), viii+133pp.Google Scholar
  2. [AW00]
    S.J. Altschuler, L.F. Wu, On deforming confoliations. J. Differ. Geom. 54, 75–97 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  3. [BBCG10]
    A. Bahri, M. Bendersky, F.R. Cohen, S. Gitler, The polyhedral product functor: a method of decomposition for moment-angle complexes, arrangements and related spaces. Adv. Math. 225(3), 1634–1668 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  4. [BV14]
    Y. Barreto, A. Verjovsky, Moment-angle manifolds, intersection of quadrics and higher dimensional contact manifolds. Moscow Math. J. 14(4), 669–696 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  5. [BLV]
    Y. Barreto, S. López de Medrano, A. Verjovsky, Open book structures on moment-angle manifolds \(Z^{\mathbb C}(\Lambda )\) and higher dimensional contact manifolds. arXiv:1303.2671Google Scholar
  6. [BLV17]
    Y. Barreto, S. López de Medrano, A. Verjovsky, Some open book and contact structures on moment-angle manifolds. Bol. Soc. Mat. Mex. 23(1), 423–437 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  7. [BHPV04]
    W. Barth, K. Hulek, C. Peters, A. Van de Ven, Compact Complex Surfaces (Springer, Berlin, 2004)zbMATHCrossRefGoogle Scholar
  8. [BA03]
    I.V. Baskakov, Massey triple products in the cohomology of moment-angle complexes. Russ. Math. Surv. 58, 1039–1041 (2003)zbMATHCrossRefGoogle Scholar
  9. [BP01]
    F. Battaglia, E. Prato, Generalized toric varieties for simple nonrational convex polytopes. Int. Math. Res. Not. 24, 1315–1337 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  10. [BA01]
    F. Battaglia, E. Prato, Simple nonrational convex polytopes via symplectic geometry. Topology 40, 961–975 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  11. [BP01]
    F. Battaglia, E. Prato Generalized toric varieties for simple nonrational convex polytopes. Int. Math. Res. Not. 24, 1315–1337 (2001)Google Scholar
  12. [BZ15]
    F. Battaglia, D. Zaffran, Foliations modeling nonrational simplicial toric varieties. Int. Math. Res. Not. IMRN 2015(22), 11785–11815 (2015)MathSciNetzbMATHGoogle Scholar
  13. [BA13]
    L. Battisti, LVMB manifolds and quotients of toric varieties. Math. Z. 275(1–2), 549–568 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  14. [BEM15]
    M.S. Borman, Y. Eliashberg, E. Murphy, Existence and classification of overtwisted contact structures in all dimensions. Acta Math. 215, 281–361 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  15. [BO01]
    F. Bosio, Variétés complexes compactes: une généralisation de la construction de Meersseman et López de Medrano-Verjovsky. Ann. Inst. Fourier 51(5), 1259–1297 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  16. [BM06]
    F. Bosio, L. Meersseman, Real quadrics in \(\mathbb {C}^n\), complex manifolds and convex polytopes. Acta Math. 197, 53–127 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  17. [BO02]
    F. Bourgeois, Odd dimensional tori are contact manifolds. Int. Math. Res. Not. 30, 1571–1574 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  18. [BP02]
    V.M. Buchstaber, T.E. Panov, Torus Actions and Their Applications in Topology and Combinatorics (AMS, Providence, 2002)Google Scholar
  19. [CE53]
    E. Calabi, B. Eckmann, A class of compact, complex manifolds which are not algebraic. Ann. Math. 58, 494–500 (1953)MathSciNetzbMATHCrossRefGoogle Scholar
  20. [CO85]
    A. Connes, Non-commutative differential geometry. Publ. Math. l’IHES 62(1), 41–144 (1985)zbMATHCrossRefGoogle Scholar
  21. [CO94]
    A. Connes, Noncommutative Geometry (Academic Press, San Diego, 1994) xiv+661pp.Google Scholar
  22. [CO95]
    D. Cox, The homogeneous coordinate ring of a toric variety. J. Algebraic Geom. 4, 17–50 (1995)MathSciNetzbMATHGoogle Scholar
  23. [CZ07]
    S. Cupit-Foutu, D. Zaffran, Non-Kähler manifolds and GIT quotients. Math. Z. 257, 783–797 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  24. [DJ91]
    M. Davis, T. Januszkiewicz Convex polytopes, coxeter orbifolds and torus actions. Duke Math. J. 62(2), 417–451 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  25. [DG17]
    J.P. Demailly, H. Gaussier, Algebraic embeddings of smooth almost complex structures. J. Eur. Math. Soc. 19(11), 3391–3419 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  26. [DM88]
    S.L. de Medrano, The Space of Siegel Leaves of a Holomorphic Vector Field. Lecture Notes in Mathematics, vol. 1345 (Springer, Berlin, 1988), pp. 233–245Google Scholar
  27. [DM89]
    S.L. de Medrano, The Topology of the Intersection of Quadrics in \(\mathbb {R}^n\). Lecture Notes in Mathematics, vol. 1370 (Springer, Berlin, 1989), pp. 280–292Google Scholar
  28. [DM14]
    S.L. de Medrano, Singularities of homogeneous quadratic mappings. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. 108(1), 95–112 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  29. [DM17]
    S.L. de Medrano, Samuel Gitler and the topology of intersections of quadrics. Bol. Soc. Mat. Mex. 23(1), 5–21 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  30. [DV97]
    S.L. de Medrano, A. Verjovsky, A new family of complex, compact, non-symplectic manifolds. Bull. Braz. Math. Soc. 28(2), 253–269 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  31. [EL90]
    Y. Eliashberg, Topological characterization of Stein manifolds of dimension > 2. Int. J. Math. 1(1), 29–46 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  32. [FU93]
    W. Fulton, Introduction to Toric Varieties (Princeton University Press, Princeton, 1993)zbMATHCrossRefGoogle Scholar
  33. [GHS83]
    J. Girbau, A. Haefliger, D. Sundararaman, On deformations of transversely holomorphic foliations. J. Reine Angew. Math. 345, 122–147 (1983)MathSciNetzbMATHGoogle Scholar
  34. [GI]
    E. Giroux, Geometrie de contact: de la dimension trois vers les dimensions superieures. Proc. Int. Congress Math. II, 405–414 (2002)Google Scholar
  35. [GL13]
    S. Gitler, S. López de Medrano, Intersections of quadrics, moment-angle manifolds and connected sums. Geom. Topol. 17(3), 1497–1534 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  36. [GL71]
    R. Goldstein, L. Lininger, A Classification of 6-Manifolds with FreeS 1 -Action. Lecture Notes in Mathematics, vol. 298 (Springer, Berlin, 1971), pp. 316–323Google Scholar
  37. [GL05]
    V. Gómez Gutiérrez, S. Lépez de Medrano, Stably Parallelizable Compact Manifolds are Complete Intersections of Quadrics. Publicaciones Preliminares del Instituto de Matemáticas (UNAM, Mxico, 2004)Google Scholar
  38. [GL14]
    G.V. Gómez, S. López de Medrano, Topology of the intersections of quadrics II. Bol. Soc. Mat. Mex. 20(2), 237–255 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  39. [GH78]
    P. Griffiths, J. Harris, Principles of Algebraic Geometry. Pure and Applied Mathematics (Wiley, New York, 1978), xii+813pp.Google Scholar
  40. [HA85]
    A. Haefliger, Deformations of transversely holomorphic flows on spheres and deformations of Hopf manifolds. Compos. Math. 55, 241–251 (1985)MathSciNetzbMATHGoogle Scholar
  41. [HI62]
    M.W. Hirsch, Smooth regular neighborhoods. Ann. Math. 76(3), 524–530 (1962)MathSciNetzbMATHCrossRefGoogle Scholar
  42. [IS0]
    H. Ishida, Towards transverse toric geometry. arXiv:1807.10449Google Scholar
  43. [IS17]
    H. Ishida, Torus invariant transverse Kähler foliations. Trans. Am. Math. Soc. 369(7), 5137–5155 (2017)zbMATHCrossRefGoogle Scholar
  44. [KA86]
    M. Kato, A. Yamada, Examples of simply connected compact complex 3-folds II. Tokyo J. Math. 9, 1–28 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  45. [KLMV14]
    L. Katzarkov, E. Lupercio, L. Meersseman, A. Verjovsky, The definition of a non-commutative toric variety, in Algebraic Topology: Applications and New Directions. Contemporary Mathematics, vol. 620 (American Mathematical Society, Providence, 2014), pp. 223–250Google Scholar
  46. [KO64]
    K. Kopfermann, Maximale Untergruppen Abelscher komplexer Liescher Gruppen. Schr. Math. Inst. Univ. Münster 29, iii+72pp. (1964)Google Scholar
  47. [LM02]
    F. Lescure, L. Meersseman, Compactifications équivariantes non kählériennes d’un groupe algébrique multiplicatif. Ann. Inst. Fourier 52, 255–273 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  48. [LN96]
    J.J. Loeb, M. Nicolau, Holomorphic flows and complex structures on products of odd-dimensional spheres. Math. Ann. 306, 781–817 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  49. [LN99]
    J.J. Loeb, M. Nicolau, On the complex geometry of a class of non-Kählerian manifolds. Israel J. Math. 110, 371–379 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  50. [MA93]
    P. MacMullen, On simple polytopes. Invent. Math. 113, 419–444 (1993)MathSciNetCrossRefGoogle Scholar
  51. [MA74]
    H. Maeda, Some complex structures on the product of spheres. J. Fac. Sci. Univ. Tokyo 21, 161–165 (1974)MathSciNetzbMATHGoogle Scholar
  52. [MC91]
    D. McDuff, Symplectic manifolds with contact type boundaries. Invent. Math. 103(3), 651–671 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  53. [MC79]
    D. McGavran, Adjacent connected sums and torus actions. Trans. Am. Math. Soc. 251, 235–254 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  54. [ME82]
    C. Meckert, Forme de contact sur la somme connexe de deux variétés de contact de dimension impare. Ann. L’Institut Fourier 32(3), 251–260 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  55. [ME98]
    L. Meersseman, Un nouveau procédé de construction géométrique de variétés compactes, complexes, non algébriques, en dimension quelconque. Ph.D. Thesis, Lille (1998)Google Scholar
  56. [ME00]
    L. Meersseman, A new geometric construction of compact complex manifolds in any dimension. Math. Ann. 317, 79–115 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  57. [MV04]
    L. Meersseman, A. Verjovsky, Holomorphic principal bundles over projective toric varieties. J. für die Reine und Angewandte Math. 572, 57–96 (2004)MathSciNetzbMATHGoogle Scholar
  58. [MV08]
    L. Meersseman, A. Verjovsky, Sur les variétés LV-M (French). [On LV-M manifolds] Singularities II. Contemporary Mathematics, vol. 475 (American Mathematical Society, Providence, 2008), pp. 111–134Google Scholar
  59. [MI65]
    J. Milnor, Lectures on the H-Cobordism Theorem (Princeton University Press, Princeton, 1965)zbMATHCrossRefGoogle Scholar
  60. [MO66]
    A. Morimoto, On the classification of non compact complex abelian Lie groups. Trans. Am. Math. Soc. 123, 200–228 (1966)zbMATHCrossRefGoogle Scholar
  61. [OR72]
    P. Orlik, Seifert Manifolds. Lecture Notes in Mathematics, vol. 291 (Springer, Berlin, 1972)zbMATHCrossRefGoogle Scholar
  62. [PA10]
    T. Panov, Moment-Angle Manifolds and Complexes. Lecture notes KAIST’2010. Taras Panov. Trends in Mathematics - New Series. ICMS, KAIST, vol. 12, no. 1 (2010), pp. 43–69Google Scholar
  63. [PR01]
    E. Prato, Simple non-rational convex polytopes via symplectic geometry. Topology 40(5), 961–975 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  64. [SC61]
    G. Scheja, Riemannsche Hebbarkeitssätze für Cohomologieklassen. Math. Ann. 144, 345–360 (1961)Google Scholar
  65. [SE11]
    P. Seidel, Simple examples of distinct Liouville type symplectic structures. J. Topol. Anal. 3(1), 1–5 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  66. [ST96]
    P.R. Stanley, Combinatorics and Commutative Algebra. Progress in Mathematics, 2nd edn., vol. 41 (Birkhäuser, Boston, 1996)Google Scholar
  67. [ST51]
    N. Steenrod, The Topology of Fibre Bundles (Princeton University Press, Princeton, 1951)zbMATHCrossRefGoogle Scholar
  68. [ST83]
    S. Sternberg, Lectures on Differential Geometry, 2nd edn. With an appendix by Sternberg and Victor W. Guillemin (Chelsea Publishing, New York, 1983)zbMATHGoogle Scholar
  69. [TA12]
    J. Tambour, LVMB manifolds and simplicial spheres. Ann. Inst. Fourier 62(4), 1289–1317 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  70. [TI99]
    V.A. Timorin, An analogue of the Hodge-Riemann relations for simple convex polytopes. Russ. Math. Surv. 54, 381–426 (1999)zbMATHCrossRefGoogle Scholar
  71. [WA80]
    C.T.C. Wall, Stability, pencils and polytopes. Bull. Lond. Math. Soc. 12, 401–421 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  72. [WE91]
    A. Weinstein, Contact surgery and symplectic handlebodies. Hokkaido Math. J. 620(2), 241–251 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  73. [WE73]
    R.O. Wells Jr., Differential Analysis on Complex Manifolds (Upper Saddle River, Prentice Hall, 1973)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Instituto de MatemáticasUniversidad Nacional Autónoma de México (UNAM)México CityMexico

Personalised recommendations